Conjugate gradients squared method


x = cgs(A,b)
[x,flag] = cgs(A,b,...)
[x,flag,relres] = cgs(A,b,...)
[x,flag,relres,iter] = cgs(A,b,...)
[x,flag,relres,iter,resvec] = cgs(A,b,...)


x = cgs(A,b) attempts to solve the system of linear equations A*x = b for x. The n-by-n coefficient matrix A must be square and should be large and sparse. The column vector b must have length n. You can specify A as a function handle, afun, such that afun(x) returns A*x.

Parameterizing Functions explains how to provide additional parameters to the function afun, as well as the preconditioner function mfun described below, if necessary.

If cgs converges, a message to that effect is displayed. If cgs fails to converge after the maximum number of iterations or halts for any reason, a warning message is printed displaying the relative residual norm(b-A*x)/norm(b) and the iteration number at which the method stopped or failed.

cgs(A,b,tol) specifies the tolerance of the method, tol. If tol is [], then cgs uses the default, 1e-6.

cgs(A,b,tol,maxit) specifies the maximum number of iterations, maxit. If maxit is [] then cgs uses the default, min(n,20).

cgs(A,b,tol,maxit,M) and cgs(A,b,tol,maxit,M1,M2) use the preconditioner M or M = M1*M2 and effectively solve the system inv(M)*A*x = inv(M)*b for x. If M is [] then cgs applies no preconditioner. M can be a function handle mfun such that mfun(x) returns M\x.

cgs(A,b,tol,maxit,M1,M2,x0) specifies the initial guess x0. If x0 is [], then cgs uses the default, an all-zero vector.

[x,flag] = cgs(A,b,...) returns a solution x and a flag that describes the convergence of cgs.




cgs converged to the desired tolerance tol within maxititerations.


cgs iterated maxit times but did not converge.


Preconditioner M was ill-conditioned.


cgs stagnated. (Two consecutive iterates were the same.)


One of the scalar quantities calculated during cgs became too small or too large to continue computing.

Whenever flag is not 0, the solution x returned is that with minimal norm residual computed over all the iterations. No messages are displayed if the flag output is specified.

[x,flag,relres] = cgs(A,b,...) also returns the relative residual norm(b-A*x)/norm(b). If flag is 0, then relres <= tol.

[x,flag,relres,iter] = cgs(A,b,...) also returns the iteration number at which x was computed, where 0 <= iter <= maxit.

[x,flag,relres,iter,resvec] = cgs(A,b,...) also returns a vector of the residual norms at each iteration, including norm(b-A*x0).


Using cgs with a Matrix Input

A = gallery('wilk',21);
b = sum(A,2);
tol = 1e-12;  maxit = 15; 
M1 = diag([10:-1:1 1 1:10]);
x = cgs(A,b,tol,maxit,M1);

displays the message

cgs converged at iteration 13 to a solution with 
relative residual 2.4e-016.

Using cgs with a Function Handle

This example replaces the matrix A in the previous example with a handle to a matrix-vector product function afun, and the preconditioner M1 with a handle to a backsolve function mfun. The example is contained in the file run_cgs that

  • Calls cgs with the function handle @afun as its first argument.

  • Contains afun as a nested function, so that all variables in run_cgs are available to afun and myfun.

The following shows the code for run_cgs:

function x1 = run_cgs
n = 21;
b = afun(ones(n,1));
tol = 1e-12;  maxit = 15; 
x1 = cgs(@afun,b,tol,maxit,@mfun);
    function y = afun(x)
        y = [0; x(1:n-1)] + ...
              [((n-1)/2:-1:0)'; (1:(n-1)/2)'].*x + ...
              [x(2:n); 0];
    function y = mfun(r)
        y = r ./ [((n-1)/2:-1:1)'; 1; (1:(n-1)/2)'];

When you enter

x1 = run_cgs

MATLAB® software returns

cgs converged at iteration 13 to a solution with 
relative residual 2.4e-016.

Using cgs with a Preconditioner.

This example demonstrates the use of a preconditioner.

Load west0479, a real 479-by-479 nonsymmetric sparse matrix.

load west0479;
A = west0479;

Define b so that the true solution is a vector of all ones.

b = full(sum(A,2));

Set the tolerance and maximum number of iterations.

tol = 1e-12;
maxit = 20;

Use cgs to find a solution at the requested tolerance and number of iterations.

[x0,fl0,rr0,it0,rv0] = cgs(A,b,tol,maxit);

fl0 is 1 because cgs does not converge to the requested tolerance 1e-12 within the requested 20 iterations. In fact, the behavior of cgs is so poor that the initial guess (x0 = zeros(size(A,2),1) is the best solution and is returned as indicated by it0 = 0. MATLAB stores the residual history in rv0.

Plot the behavior of cgs.

xlabel('Iteration number');
ylabel('Relative residual');

The plot shows that the solution does not converge. You can use a preconditioner to improve the outcome.

Create a preconditioner with ilu, since A is nonsymmetric.

[L,U] = ilu(A,struct('type','ilutp','droptol',1e-5));
Error using ilu
There is a pivot equal to zero.  Consider decreasing the 
drop tolerance or consider using the 'udiag' option.

MATLAB cannot construct the incomplete LU as it would result in a singular factor, which is useless as a preconditioner.

You can try again with a reduced drop tolerance, as indicated by the error message.

[L,U] = ilu(A,struct('type','ilutp','droptol',1e-6));
[x1,fl1,rr1,it1,rv1] = cgs(A,b,tol,maxit,L,U);

fl1 is 0 because cgs drives the relative residual to 4.3851e-014 (the value of rr1). The relative residual is less than the prescribed tolerance of 1e-12 at the third iteration (the value of it1) when preconditioned by the incomplete LU factorization with a drop tolerance of 1e-6. The output rv1(1) is norm(b) and the output rv1(14) is norm(b-A*x2).

You can follow the progress of cgs by plotting the relative residuals at each iteration starting from the initial estimate (iterate number 0).

xlabel('Iteration number');
ylabel('Relative residual');


[1] Barrett, R., M. Berry, T. F. Chan, et al., Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods, SIAM, Philadelphia, 1994.

[2] Sonneveld, Peter, “CGS: A fast Lanczos-type solver for nonsymmetric linear systems,” SIAM J. Sci. Stat. Comput., January 1989, Vol. 10, No. 1, pp. 36–52.

Extended Capabilities

Introduced before R2006a